Logical seduction and historical delusion

John Edward Terrell

Please note: this commentary, recovered on 9-Jan-2017, was originally published in Science Dialogues on 20-Feb-2015.

In his acclaimed novel The Oxford Murders, the Argentinean writer and mathematician Guillermo Martínez engagingly shows how easy it is to hide the truth from others by getting them to think that a series of similar events—in this instance, a series of murders—is happening because, when taken in sequence, they appear to add up to a coded message that we are being taunted to decipher.

Judging by appearances, each murder apparently symbolizes one of the logical steps in a predictable sequence, just as most of us would probably agree that the next logical number in the familiar series 2, 4, 8, and 16 must be the multiple 32. Perhaps, but as the philosopher Ludwig Wittgenstein famously observed, any finite sequence of numbers can be continued in a variety of different ways, not just in the one way that may seem reasonable (Biletzki and Matar 2006).

For example, the narrator, whose name we are never told, is asked early in this novel if he can figure out what is the next symbol in the odd series reproduced here as Fig. 1a.

Although Martínez never shows us the solution he has in mind (the narrator merely tells us later on that the answer is the number series 1, 2, 3, 4), we suspect those who find riddles like this one appealing are likely to say the solution shown in Fig. 1b is the right resolve: an answer derived from the rules of symmetry (Fig. 1c). Yet in keeping with Martínez’s revealing observations about both logic and magic set here and there in this story, what if the proper solution is not so playful?

For example, what if the three symbols already revealed follow instead the alternative rule that one stroke equals 1? If this were so, then the missing fourth symbol in this cryptic series would not be an “M” with a bar drawn horizontally through it (in keeping with our different rule, this strange symbol could stand instead for the number 5), but disconcertingly could be drawn either as a single stroke (Fig. 1d), or possibly as an inscribed circle, the letter “O,” or a zero (Fig. 1e).

Doubt as to the proper resolve of Martínez’s series of symbols illustrates Wittgenstein’s cryptic and oft-quoted remark: “This was our paradox: no course of action could be determined by a rule, because every course of action can be made out to accord with the rule. The answer was: if everything can be made out to accord with the rule, then it can also be made out to conflict with it. And so there would be neither accord nor conflict” (quoted in: Biletzki and Matar 2006).

I am not a philosopher, nor a novelist. It seems to me, however, that Martinez’s tale and Wittgenstein’s remark both tell us something about ourselves, about how we are given to looking for similarities among things and events proving that what we are seeing makes sense not by chance but necessity. It might even be argued that human beings are strongly predisposed to equate similarity with necessity.

This is why we need statisticians, however much statistics may sometimes seem only a cultivated way of lying for effect. They keep us from foolishly jumping to the conclusion that similarities in appearance or similarities in effect are necessarily similarities of cause.

And in this regard, we need to remember that when statisticians say that something should be attributed to “chance,” they do not mean “without cause.” Far from it: the point they are making is that the cause (or causes) is not necessarily the one we think it is.